## Friday, 9 August 2013

### Are there simple models for working out when a population will go extinct?

I recently received this from Dr John Lewis, and old friend of mine, when I asked him whether I'd get censured for that tweet I posted on @paulakahumbu, below:

What's wrong here, please? Population 450,000/ Killed pday 120 / Days in yr 365 = 10 yrs extinct

Seems from the '@toptweet' bumperdinkle there it became a subject of quite a lot of discussion.  Well, I was trying to say, "How is it possible for the African elephant to go extinct if this assumes that 120 elephants are being slaughtered everyday?"  Where is the reality?  Paula suggested, at the museum, that about 3 are being killed per day and the figure is rising; then Ol Pejeta came back to me and told me to look again and the figure rose to a whopping 30 per day....  I still can't quite wrap my mind around this.  But for them to go extinct in 10 years we are talking bigtime.  When Amanpour can report that they are using helicopters and machine guns, etcetera, and, now that it seems that poaching has turned into genocide, obviously, we are bound to be reeling!  120 is not difficult for them, evidently.  I felt something was wrong with the math but they are scientists.  They must have done a lot of homework before publicizing the 10 year cut off point.

Unfortunately, you might be censured for that. ;-) The following is an attempt to describe how the prediction could be done.

The change in the total population each day is the number born minus the number who die by being killed and by other means. However, all three of those numbers depend on the population, and specifically on the population distribution against age, and on the rate at which each of those events occur against age.

In particular, the birth rate depends on the rate of mating at a time the gestation period earlier, and the success rate; and the rate of mating depends on the number of each sex against age, and the probability of them mating against age.

As the population is spread over an area, many of these parameters might depend on the habitat and the numbers in each type of habitat, and the rate of mating also depends on the rate at which they meet, etc..

No doubt, there is also seasonal behaviour, including migration, to be included.

Put that lot together and there is a set of differential equations to solve with some parameters and some initial conditions representing the situation at some starting point in time.

Although it is complex, there is nothing particularly mysterious about any of that; the difficulty is that assumptions need to be made about the parameters and their dependence on age, habitat, time and, possibly on population itself.

People are doing this kind of modelling, such as these: